MONOTONICITY RESULTS OF SOLUTIONS FOR SYSTEM INVOLVING THE FRACTIONAL p-LAPLACIAN

In this paper, we investigate the monotonicity of solutions to the fractional p-Laplacian system given by (-triangle)(p1)(s1) u(x) = f (u(x), v(x)), x is an element of ohm, (-triangle)(p2)(s2 ) v(x) = g(u(x), v(x)), x is an element of ohm, where 0 < s(1), s(2) < 1, 2 <= p(1), p(2) < infinity, and n > 2. The domain ohm subset of R-n is either a bounded domain that is convex in xn-direction, an unbounded domain, or the whole space. The operator (-triangle)(p)(s) denotes the fractional p-Laplacian (with s = s(1), s(2) and p = p(1), p(2)), and f,g is an element of C-1(R-2). Under suitable conditions on the functions f and g, we employ the sliding method to demonstrate that any solution (u, v) of the system is strictly increasing in ohm with respect to x(n). The proof involves the idea of singular integral estimation along a sequence of approximate maximum points, which was originally introduced by Wu and Chen [22].